题目内容
椭圆
=1(a>b>0)的右焦点F,其右准线与x轴的交点为A,在椭圆上存在点P满足线段AP的垂直平分线过点F,则椭圆离心率的取值范围是________.
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解法1)由题意,椭圆上存在点P,使得线段AP的垂直平分线过点F,所以|PF|=|FA|,而|FA|=
-c,|PF|≤a+c,所以
-c≤a+c,即a2≤ac+2c2.又e=
,所以2e2+e≥1,所以2e2+e-1≥0,即(2e-1)(e+1)≥0.又0<e<1,所以
≤e<1.
(解法2)设点P(x,y).由题意,椭圆上存在点P,使得线段AP的垂直平分线过点F,所以|PF|=|FA|,由椭圆第二定义,
=e,所以|PF|=
e-ex=a-ex,而|FA|=
-c,所以a-ex=
-c,解得x=
(a+c-
).由于-a≤x≤a,
所以-a≤
(a+c-
)≤a.又e=
,所以2e2+e-1≥0,即(2e-1)(e+1)≥0.
又0<e<1,所以
≤e<1.
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(解法2)设点P(x,y).由题意,椭圆上存在点P,使得线段AP的垂直平分线过点F,所以|PF|=|FA|,由椭圆第二定义,
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所以-a≤
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又0<e<1,所以
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